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Theorem madebdayim 27382
Description: If a surreal is a member of a made set, its birthday is less than or equal to the level. (Contributed by Scott Fenton, 10-Aug-2024.)
Assertion
Ref Expression
madebdayim (𝑋 ∈ ( M β€˜π΄) β†’ ( bday β€˜π‘‹) βŠ† 𝐴)

Proof of Theorem madebdayim
Dummy variables π‘Ž 𝑏 π‘₯ 𝑦 𝑧 𝑙 π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvdm 6929 . . 3 (𝑋 ∈ ( M β€˜π΄) β†’ 𝐴 ∈ dom M )
2 madef 27351 . . . 4 M :OnβŸΆπ’« No
32fdmi 6730 . . 3 dom M = On
41, 3eleqtrdi 2844 . 2 (𝑋 ∈ ( M β€˜π΄) β†’ 𝐴 ∈ On)
5 fveq2 6892 . . . . . 6 (π‘Ž = 𝑏 β†’ ( M β€˜π‘Ž) = ( M β€˜π‘))
6 sseq2 4009 . . . . . 6 (π‘Ž = 𝑏 β†’ (( bday β€˜π‘₯) βŠ† π‘Ž ↔ ( bday β€˜π‘₯) βŠ† 𝑏))
75, 6raleqbidv 3343 . . . . 5 (π‘Ž = 𝑏 β†’ (βˆ€π‘₯ ∈ ( M β€˜π‘Ž)( bday β€˜π‘₯) βŠ† π‘Ž ↔ βˆ€π‘₯ ∈ ( M β€˜π‘)( bday β€˜π‘₯) βŠ† 𝑏))
8 fveq2 6892 . . . . . . 7 (π‘₯ = 𝑦 β†’ ( bday β€˜π‘₯) = ( bday β€˜π‘¦))
98sseq1d 4014 . . . . . 6 (π‘₯ = 𝑦 β†’ (( bday β€˜π‘₯) βŠ† 𝑏 ↔ ( bday β€˜π‘¦) βŠ† 𝑏))
109cbvralvw 3235 . . . . 5 (βˆ€π‘₯ ∈ ( M β€˜π‘)( bday β€˜π‘₯) βŠ† 𝑏 ↔ βˆ€π‘¦ ∈ ( M β€˜π‘)( bday β€˜π‘¦) βŠ† 𝑏)
117, 10bitrdi 287 . . . 4 (π‘Ž = 𝑏 β†’ (βˆ€π‘₯ ∈ ( M β€˜π‘Ž)( bday β€˜π‘₯) βŠ† π‘Ž ↔ βˆ€π‘¦ ∈ ( M β€˜π‘)( bday β€˜π‘¦) βŠ† 𝑏))
12 fveq2 6892 . . . . 5 (π‘Ž = 𝐴 β†’ ( M β€˜π‘Ž) = ( M β€˜π΄))
13 sseq2 4009 . . . . 5 (π‘Ž = 𝐴 β†’ (( bday β€˜π‘₯) βŠ† π‘Ž ↔ ( bday β€˜π‘₯) βŠ† 𝐴))
1412, 13raleqbidv 3343 . . . 4 (π‘Ž = 𝐴 β†’ (βˆ€π‘₯ ∈ ( M β€˜π‘Ž)( bday β€˜π‘₯) βŠ† π‘Ž ↔ βˆ€π‘₯ ∈ ( M β€˜π΄)( bday β€˜π‘₯) βŠ† 𝐴))
15 elmade2 27363 . . . . . . . 8 (π‘Ž ∈ On β†’ (π‘₯ ∈ ( M β€˜π‘Ž) ↔ βˆƒπ‘™ ∈ 𝒫 ( O β€˜π‘Ž)βˆƒπ‘Ÿ ∈ 𝒫 ( O β€˜π‘Ž)(𝑙 <<s π‘Ÿ ∧ (𝑙 |s π‘Ÿ) = π‘₯)))
1615adantr 482 . . . . . . 7 ((π‘Ž ∈ On ∧ βˆ€π‘ ∈ π‘Ž βˆ€π‘¦ ∈ ( M β€˜π‘)( bday β€˜π‘¦) βŠ† 𝑏) β†’ (π‘₯ ∈ ( M β€˜π‘Ž) ↔ βˆƒπ‘™ ∈ 𝒫 ( O β€˜π‘Ž)βˆƒπ‘Ÿ ∈ 𝒫 ( O β€˜π‘Ž)(𝑙 <<s π‘Ÿ ∧ (𝑙 |s π‘Ÿ) = π‘₯)))
17 elpwi 4610 . . . . . . . . . . 11 (𝑙 ∈ 𝒫 ( O β€˜π‘Ž) β†’ 𝑙 βŠ† ( O β€˜π‘Ž))
18 elpwi 4610 . . . . . . . . . . 11 (π‘Ÿ ∈ 𝒫 ( O β€˜π‘Ž) β†’ π‘Ÿ βŠ† ( O β€˜π‘Ž))
1917, 18anim12i 614 . . . . . . . . . 10 ((𝑙 ∈ 𝒫 ( O β€˜π‘Ž) ∧ π‘Ÿ ∈ 𝒫 ( O β€˜π‘Ž)) β†’ (𝑙 βŠ† ( O β€˜π‘Ž) ∧ π‘Ÿ βŠ† ( O β€˜π‘Ž)))
20 unss 4185 . . . . . . . . . 10 ((𝑙 βŠ† ( O β€˜π‘Ž) ∧ π‘Ÿ βŠ† ( O β€˜π‘Ž)) ↔ (𝑙 βˆͺ π‘Ÿ) βŠ† ( O β€˜π‘Ž))
2119, 20sylib 217 . . . . . . . . 9 ((𝑙 ∈ 𝒫 ( O β€˜π‘Ž) ∧ π‘Ÿ ∈ 𝒫 ( O β€˜π‘Ž)) β†’ (𝑙 βˆͺ π‘Ÿ) βŠ† ( O β€˜π‘Ž))
22 simpr 486 . . . . . . . . . . . . 13 ((((π‘Ž ∈ On ∧ βˆ€π‘ ∈ π‘Ž βˆ€π‘¦ ∈ ( M β€˜π‘)( bday β€˜π‘¦) βŠ† 𝑏) ∧ (𝑙 βˆͺ π‘Ÿ) βŠ† ( O β€˜π‘Ž)) ∧ 𝑙 <<s π‘Ÿ) β†’ 𝑙 <<s π‘Ÿ)
23 simplll 774 . . . . . . . . . . . . 13 ((((π‘Ž ∈ On ∧ βˆ€π‘ ∈ π‘Ž βˆ€π‘¦ ∈ ( M β€˜π‘)( bday β€˜π‘¦) βŠ† 𝑏) ∧ (𝑙 βˆͺ π‘Ÿ) βŠ† ( O β€˜π‘Ž)) ∧ 𝑙 <<s π‘Ÿ) β†’ π‘Ž ∈ On)
24 dfss3 3971 . . . . . . . . . . . . . . . . 17 ((𝑙 βˆͺ π‘Ÿ) βŠ† ( O β€˜π‘Ž) ↔ βˆ€π‘§ ∈ (𝑙 βˆͺ π‘Ÿ)𝑧 ∈ ( O β€˜π‘Ž))
25 fveq2 6892 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑧 β†’ ( bday β€˜π‘¦) = ( bday β€˜π‘§))
2625sseq1d 4014 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = 𝑧 β†’ (( bday β€˜π‘¦) βŠ† 𝑏 ↔ ( bday β€˜π‘§) βŠ† 𝑏))
2726rspccv 3610 . . . . . . . . . . . . . . . . . . . . . 22 (βˆ€π‘¦ ∈ ( M β€˜π‘)( bday β€˜π‘¦) βŠ† 𝑏 β†’ (𝑧 ∈ ( M β€˜π‘) β†’ ( bday β€˜π‘§) βŠ† 𝑏))
2827ralimi 3084 . . . . . . . . . . . . . . . . . . . . 21 (βˆ€π‘ ∈ π‘Ž βˆ€π‘¦ ∈ ( M β€˜π‘)( bday β€˜π‘¦) βŠ† 𝑏 β†’ βˆ€π‘ ∈ π‘Ž (𝑧 ∈ ( M β€˜π‘) β†’ ( bday β€˜π‘§) βŠ† 𝑏))
29 rexim 3088 . . . . . . . . . . . . . . . . . . . . 21 (βˆ€π‘ ∈ π‘Ž (𝑧 ∈ ( M β€˜π‘) β†’ ( bday β€˜π‘§) βŠ† 𝑏) β†’ (βˆƒπ‘ ∈ π‘Ž 𝑧 ∈ ( M β€˜π‘) β†’ βˆƒπ‘ ∈ π‘Ž ( bday β€˜π‘§) βŠ† 𝑏))
3028, 29syl 17 . . . . . . . . . . . . . . . . . . . 20 (βˆ€π‘ ∈ π‘Ž βˆ€π‘¦ ∈ ( M β€˜π‘)( bday β€˜π‘¦) βŠ† 𝑏 β†’ (βˆƒπ‘ ∈ π‘Ž 𝑧 ∈ ( M β€˜π‘) β†’ βˆƒπ‘ ∈ π‘Ž ( bday β€˜π‘§) βŠ† 𝑏))
3130adantl 483 . . . . . . . . . . . . . . . . . . 19 ((π‘Ž ∈ On ∧ βˆ€π‘ ∈ π‘Ž βˆ€π‘¦ ∈ ( M β€˜π‘)( bday β€˜π‘¦) βŠ† 𝑏) β†’ (βˆƒπ‘ ∈ π‘Ž 𝑧 ∈ ( M β€˜π‘) β†’ βˆƒπ‘ ∈ π‘Ž ( bday β€˜π‘§) βŠ† 𝑏))
32 elold 27364 . . . . . . . . . . . . . . . . . . . 20 (π‘Ž ∈ On β†’ (𝑧 ∈ ( O β€˜π‘Ž) ↔ βˆƒπ‘ ∈ π‘Ž 𝑧 ∈ ( M β€˜π‘)))
3332adantr 482 . . . . . . . . . . . . . . . . . . 19 ((π‘Ž ∈ On ∧ βˆ€π‘ ∈ π‘Ž βˆ€π‘¦ ∈ ( M β€˜π‘)( bday β€˜π‘¦) βŠ† 𝑏) β†’ (𝑧 ∈ ( O β€˜π‘Ž) ↔ βˆƒπ‘ ∈ π‘Ž 𝑧 ∈ ( M β€˜π‘)))
34 bdayelon 27278 . . . . . . . . . . . . . . . . . . . . 21 ( bday β€˜π‘§) ∈ On
35 onelssex 6413 . . . . . . . . . . . . . . . . . . . . 21 ((( bday β€˜π‘§) ∈ On ∧ π‘Ž ∈ On) β†’ (( bday β€˜π‘§) ∈ π‘Ž ↔ βˆƒπ‘ ∈ π‘Ž ( bday β€˜π‘§) βŠ† 𝑏))
3634, 35mpan 689 . . . . . . . . . . . . . . . . . . . 20 (π‘Ž ∈ On β†’ (( bday β€˜π‘§) ∈ π‘Ž ↔ βˆƒπ‘ ∈ π‘Ž ( bday β€˜π‘§) βŠ† 𝑏))
3736adantr 482 . . . . . . . . . . . . . . . . . . 19 ((π‘Ž ∈ On ∧ βˆ€π‘ ∈ π‘Ž βˆ€π‘¦ ∈ ( M β€˜π‘)( bday β€˜π‘¦) βŠ† 𝑏) β†’ (( bday β€˜π‘§) ∈ π‘Ž ↔ βˆƒπ‘ ∈ π‘Ž ( bday β€˜π‘§) βŠ† 𝑏))
3831, 33, 373imtr4d 294 . . . . . . . . . . . . . . . . . 18 ((π‘Ž ∈ On ∧ βˆ€π‘ ∈ π‘Ž βˆ€π‘¦ ∈ ( M β€˜π‘)( bday β€˜π‘¦) βŠ† 𝑏) β†’ (𝑧 ∈ ( O β€˜π‘Ž) β†’ ( bday β€˜π‘§) ∈ π‘Ž))
3938ralimdv 3170 . . . . . . . . . . . . . . . . 17 ((π‘Ž ∈ On ∧ βˆ€π‘ ∈ π‘Ž βˆ€π‘¦ ∈ ( M β€˜π‘)( bday β€˜π‘¦) βŠ† 𝑏) β†’ (βˆ€π‘§ ∈ (𝑙 βˆͺ π‘Ÿ)𝑧 ∈ ( O β€˜π‘Ž) β†’ βˆ€π‘§ ∈ (𝑙 βˆͺ π‘Ÿ)( bday β€˜π‘§) ∈ π‘Ž))
4024, 39biimtrid 241 . . . . . . . . . . . . . . . 16 ((π‘Ž ∈ On ∧ βˆ€π‘ ∈ π‘Ž βˆ€π‘¦ ∈ ( M β€˜π‘)( bday β€˜π‘¦) βŠ† 𝑏) β†’ ((𝑙 βˆͺ π‘Ÿ) βŠ† ( O β€˜π‘Ž) β†’ βˆ€π‘§ ∈ (𝑙 βˆͺ π‘Ÿ)( bday β€˜π‘§) ∈ π‘Ž))
4140imp 408 . . . . . . . . . . . . . . 15 (((π‘Ž ∈ On ∧ βˆ€π‘ ∈ π‘Ž βˆ€π‘¦ ∈ ( M β€˜π‘)( bday β€˜π‘¦) βŠ† 𝑏) ∧ (𝑙 βˆͺ π‘Ÿ) βŠ† ( O β€˜π‘Ž)) β†’ βˆ€π‘§ ∈ (𝑙 βˆͺ π‘Ÿ)( bday β€˜π‘§) ∈ π‘Ž)
4241adantr 482 . . . . . . . . . . . . . 14 ((((π‘Ž ∈ On ∧ βˆ€π‘ ∈ π‘Ž βˆ€π‘¦ ∈ ( M β€˜π‘)( bday β€˜π‘¦) βŠ† 𝑏) ∧ (𝑙 βˆͺ π‘Ÿ) βŠ† ( O β€˜π‘Ž)) ∧ 𝑙 <<s π‘Ÿ) β†’ βˆ€π‘§ ∈ (𝑙 βˆͺ π‘Ÿ)( bday β€˜π‘§) ∈ π‘Ž)
43 bdayfun 27274 . . . . . . . . . . . . . . . . 17 Fun bday
44 oldssno 27356 . . . . . . . . . . . . . . . . . . 19 ( O β€˜π‘Ž) βŠ† No
45 sstr 3991 . . . . . . . . . . . . . . . . . . 19 (((𝑙 βˆͺ π‘Ÿ) βŠ† ( O β€˜π‘Ž) ∧ ( O β€˜π‘Ž) βŠ† No ) β†’ (𝑙 βˆͺ π‘Ÿ) βŠ† No )
4644, 45mpan2 690 . . . . . . . . . . . . . . . . . 18 ((𝑙 βˆͺ π‘Ÿ) βŠ† ( O β€˜π‘Ž) β†’ (𝑙 βˆͺ π‘Ÿ) βŠ† No )
47 bdaydm 27276 . . . . . . . . . . . . . . . . . 18 dom bday = No
4846, 47sseqtrrdi 4034 . . . . . . . . . . . . . . . . 17 ((𝑙 βˆͺ π‘Ÿ) βŠ† ( O β€˜π‘Ž) β†’ (𝑙 βˆͺ π‘Ÿ) βŠ† dom bday )
49 funimass4 6957 . . . . . . . . . . . . . . . . 17 ((Fun bday ∧ (𝑙 βˆͺ π‘Ÿ) βŠ† dom bday ) β†’ (( bday β€œ (𝑙 βˆͺ π‘Ÿ)) βŠ† π‘Ž ↔ βˆ€π‘§ ∈ (𝑙 βˆͺ π‘Ÿ)( bday β€˜π‘§) ∈ π‘Ž))
5043, 48, 49sylancr 588 . . . . . . . . . . . . . . . 16 ((𝑙 βˆͺ π‘Ÿ) βŠ† ( O β€˜π‘Ž) β†’ (( bday β€œ (𝑙 βˆͺ π‘Ÿ)) βŠ† π‘Ž ↔ βˆ€π‘§ ∈ (𝑙 βˆͺ π‘Ÿ)( bday β€˜π‘§) ∈ π‘Ž))
5150adantl 483 . . . . . . . . . . . . . . 15 (((π‘Ž ∈ On ∧ βˆ€π‘ ∈ π‘Ž βˆ€π‘¦ ∈ ( M β€˜π‘)( bday β€˜π‘¦) βŠ† 𝑏) ∧ (𝑙 βˆͺ π‘Ÿ) βŠ† ( O β€˜π‘Ž)) β†’ (( bday β€œ (𝑙 βˆͺ π‘Ÿ)) βŠ† π‘Ž ↔ βˆ€π‘§ ∈ (𝑙 βˆͺ π‘Ÿ)( bday β€˜π‘§) ∈ π‘Ž))
5251adantr 482 . . . . . . . . . . . . . 14 ((((π‘Ž ∈ On ∧ βˆ€π‘ ∈ π‘Ž βˆ€π‘¦ ∈ ( M β€˜π‘)( bday β€˜π‘¦) βŠ† 𝑏) ∧ (𝑙 βˆͺ π‘Ÿ) βŠ† ( O β€˜π‘Ž)) ∧ 𝑙 <<s π‘Ÿ) β†’ (( bday β€œ (𝑙 βˆͺ π‘Ÿ)) βŠ† π‘Ž ↔ βˆ€π‘§ ∈ (𝑙 βˆͺ π‘Ÿ)( bday β€˜π‘§) ∈ π‘Ž))
5342, 52mpbird 257 . . . . . . . . . . . . 13 ((((π‘Ž ∈ On ∧ βˆ€π‘ ∈ π‘Ž βˆ€π‘¦ ∈ ( M β€˜π‘)( bday β€˜π‘¦) βŠ† 𝑏) ∧ (𝑙 βˆͺ π‘Ÿ) βŠ† ( O β€˜π‘Ž)) ∧ 𝑙 <<s π‘Ÿ) β†’ ( bday β€œ (𝑙 βˆͺ π‘Ÿ)) βŠ† π‘Ž)
54 scutbdaybnd 27316 . . . . . . . . . . . . 13 ((𝑙 <<s π‘Ÿ ∧ π‘Ž ∈ On ∧ ( bday β€œ (𝑙 βˆͺ π‘Ÿ)) βŠ† π‘Ž) β†’ ( bday β€˜(𝑙 |s π‘Ÿ)) βŠ† π‘Ž)
5522, 23, 53, 54syl3anc 1372 . . . . . . . . . . . 12 ((((π‘Ž ∈ On ∧ βˆ€π‘ ∈ π‘Ž βˆ€π‘¦ ∈ ( M β€˜π‘)( bday β€˜π‘¦) βŠ† 𝑏) ∧ (𝑙 βˆͺ π‘Ÿ) βŠ† ( O β€˜π‘Ž)) ∧ 𝑙 <<s π‘Ÿ) β†’ ( bday β€˜(𝑙 |s π‘Ÿ)) βŠ† π‘Ž)
56 fveq2 6892 . . . . . . . . . . . . 13 ((𝑙 |s π‘Ÿ) = π‘₯ β†’ ( bday β€˜(𝑙 |s π‘Ÿ)) = ( bday β€˜π‘₯))
5756sseq1d 4014 . . . . . . . . . . . 12 ((𝑙 |s π‘Ÿ) = π‘₯ β†’ (( bday β€˜(𝑙 |s π‘Ÿ)) βŠ† π‘Ž ↔ ( bday β€˜π‘₯) βŠ† π‘Ž))
5855, 57syl5ibcom 244 . . . . . . . . . . 11 ((((π‘Ž ∈ On ∧ βˆ€π‘ ∈ π‘Ž βˆ€π‘¦ ∈ ( M β€˜π‘)( bday β€˜π‘¦) βŠ† 𝑏) ∧ (𝑙 βˆͺ π‘Ÿ) βŠ† ( O β€˜π‘Ž)) ∧ 𝑙 <<s π‘Ÿ) β†’ ((𝑙 |s π‘Ÿ) = π‘₯ β†’ ( bday β€˜π‘₯) βŠ† π‘Ž))
5958expimpd 455 . . . . . . . . . 10 (((π‘Ž ∈ On ∧ βˆ€π‘ ∈ π‘Ž βˆ€π‘¦ ∈ ( M β€˜π‘)( bday β€˜π‘¦) βŠ† 𝑏) ∧ (𝑙 βˆͺ π‘Ÿ) βŠ† ( O β€˜π‘Ž)) β†’ ((𝑙 <<s π‘Ÿ ∧ (𝑙 |s π‘Ÿ) = π‘₯) β†’ ( bday β€˜π‘₯) βŠ† π‘Ž))
6059ex 414 . . . . . . . . 9 ((π‘Ž ∈ On ∧ βˆ€π‘ ∈ π‘Ž βˆ€π‘¦ ∈ ( M β€˜π‘)( bday β€˜π‘¦) βŠ† 𝑏) β†’ ((𝑙 βˆͺ π‘Ÿ) βŠ† ( O β€˜π‘Ž) β†’ ((𝑙 <<s π‘Ÿ ∧ (𝑙 |s π‘Ÿ) = π‘₯) β†’ ( bday β€˜π‘₯) βŠ† π‘Ž)))
6121, 60syl5 34 . . . . . . . 8 ((π‘Ž ∈ On ∧ βˆ€π‘ ∈ π‘Ž βˆ€π‘¦ ∈ ( M β€˜π‘)( bday β€˜π‘¦) βŠ† 𝑏) β†’ ((𝑙 ∈ 𝒫 ( O β€˜π‘Ž) ∧ π‘Ÿ ∈ 𝒫 ( O β€˜π‘Ž)) β†’ ((𝑙 <<s π‘Ÿ ∧ (𝑙 |s π‘Ÿ) = π‘₯) β†’ ( bday β€˜π‘₯) βŠ† π‘Ž)))
6261rexlimdvv 3211 . . . . . . 7 ((π‘Ž ∈ On ∧ βˆ€π‘ ∈ π‘Ž βˆ€π‘¦ ∈ ( M β€˜π‘)( bday β€˜π‘¦) βŠ† 𝑏) β†’ (βˆƒπ‘™ ∈ 𝒫 ( O β€˜π‘Ž)βˆƒπ‘Ÿ ∈ 𝒫 ( O β€˜π‘Ž)(𝑙 <<s π‘Ÿ ∧ (𝑙 |s π‘Ÿ) = π‘₯) β†’ ( bday β€˜π‘₯) βŠ† π‘Ž))
6316, 62sylbid 239 . . . . . 6 ((π‘Ž ∈ On ∧ βˆ€π‘ ∈ π‘Ž βˆ€π‘¦ ∈ ( M β€˜π‘)( bday β€˜π‘¦) βŠ† 𝑏) β†’ (π‘₯ ∈ ( M β€˜π‘Ž) β†’ ( bday β€˜π‘₯) βŠ† π‘Ž))
6463ralrimiv 3146 . . . . 5 ((π‘Ž ∈ On ∧ βˆ€π‘ ∈ π‘Ž βˆ€π‘¦ ∈ ( M β€˜π‘)( bday β€˜π‘¦) βŠ† 𝑏) β†’ βˆ€π‘₯ ∈ ( M β€˜π‘Ž)( bday β€˜π‘₯) βŠ† π‘Ž)
6564ex 414 . . . 4 (π‘Ž ∈ On β†’ (βˆ€π‘ ∈ π‘Ž βˆ€π‘¦ ∈ ( M β€˜π‘)( bday β€˜π‘¦) βŠ† 𝑏 β†’ βˆ€π‘₯ ∈ ( M β€˜π‘Ž)( bday β€˜π‘₯) βŠ† π‘Ž))
6611, 14, 65tfis3 7847 . . 3 (𝐴 ∈ On β†’ βˆ€π‘₯ ∈ ( M β€˜π΄)( bday β€˜π‘₯) βŠ† 𝐴)
67 fveq2 6892 . . . . 5 (π‘₯ = 𝑋 β†’ ( bday β€˜π‘₯) = ( bday β€˜π‘‹))
6867sseq1d 4014 . . . 4 (π‘₯ = 𝑋 β†’ (( bday β€˜π‘₯) βŠ† 𝐴 ↔ ( bday β€˜π‘‹) βŠ† 𝐴))
6968rspccv 3610 . . 3 (βˆ€π‘₯ ∈ ( M β€˜π΄)( bday β€˜π‘₯) βŠ† 𝐴 β†’ (𝑋 ∈ ( M β€˜π΄) β†’ ( bday β€˜π‘‹) βŠ† 𝐴))
7066, 69syl 17 . 2 (𝐴 ∈ On β†’ (𝑋 ∈ ( M β€˜π΄) β†’ ( bday β€˜π‘‹) βŠ† 𝐴))
714, 70mpcom 38 1 (𝑋 ∈ ( M β€˜π΄) β†’ ( bday β€˜π‘‹) βŠ† 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071   βˆͺ cun 3947   βŠ† wss 3949  π’« cpw 4603   class class class wbr 5149  dom cdm 5677   β€œ cima 5680  Oncon0 6365  Fun wfun 6538  β€˜cfv 6544  (class class class)co 7409   No csur 27143   bday cbday 27145   <<s csslt 27282   |s cscut 27284   M cmade 27337   O cold 27338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-1o 8466  df-2o 8467  df-no 27146  df-slt 27147  df-bday 27148  df-sslt 27283  df-scut 27285  df-made 27342  df-old 27343
This theorem is referenced by:  oldbdayim  27383  madebday  27394
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